Graded n-Absorbing Ideals and their Combinatorial Structure

Graded $n$-absorbing ideals generalize graded prime ideals by extending absorption properties to products of $(n+1)$ homogeneous elements. We study several generalizations of graded prime ideals, including graded $n$-absorbing, graded weakly $n$-absorbing, graded strongly $n$-absorbing, and graded $n$-absorbing primary ideals, as well as related graded $n$-absorbing subgroups. Our primary result establishes a combinatorial model for graded $n$-absorbing principal monomial ideals in polynomial rings with the standard grading. By identifying principal monomial ideals with exponential vectors in $\mathbb{N}^{m}$, we show that the graded $n$-absorbing principal monomial ideals correspond precisely to lattice points in the simplex $\{\alpha \in \mathbb{N}^{m}: \vert \alpha \vert \leq n\}.$ Consequently, the Hasse diagram of principal monomial ideals is realized as the 1-skeleton of the Cayley graph of $\mathbb{N}^{m}$, yielding a geometric and combinatorial interpretation of graded $n$-absorption.